An equivalent of this in the multi-dimensional setting is Hanson-Wright inequality [1]}: let \(X \sim \mathcal {N}(\mu ,\Sigma )\)  for \(\Sigma \)  a positive definite matrix, \(\mu \in \mathbb {R}^d\) . Then, for any \(t>0\) ,
\(\mathbb {P}\left(\left\Vert \frac{1}{n} \sum _{i=1}^n X_i-\mu \right\Vert ^2 > \frac{2Tr(\Sigma )}{n}+\frac{9t\Vert \Sigma \Vert _{op}}{n} \right)\le e^{-t}.\) 
